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Mirrors > Home > MPE Home > Th. List > asclfval | Structured version Visualization version GIF version |
Description: Function value of the algebra scalar lifting function. (Contributed by Mario Carneiro, 8-Mar-2015.) |
Ref | Expression |
---|---|
asclfval.a | ⊢ 𝐴 = (algSc‘𝑊) |
asclfval.f | ⊢ 𝐹 = (Scalar‘𝑊) |
asclfval.k | ⊢ 𝐾 = (Base‘𝐹) |
asclfval.s | ⊢ · = ( ·𝑠 ‘𝑊) |
asclfval.o | ⊢ 1 = (1r‘𝑊) |
Ref | Expression |
---|---|
asclfval | ⊢ 𝐴 = (𝑥 ∈ 𝐾 ↦ (𝑥 · 1 )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | asclfval.a | . 2 ⊢ 𝐴 = (algSc‘𝑊) | |
2 | fveq2 6907 | . . . . . . . 8 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) = (Scalar‘𝑊)) | |
3 | asclfval.f | . . . . . . . 8 ⊢ 𝐹 = (Scalar‘𝑊) | |
4 | 2, 3 | eqtr4di 2793 | . . . . . . 7 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) = 𝐹) |
5 | 4 | fveq2d 6911 | . . . . . 6 ⊢ (𝑤 = 𝑊 → (Base‘(Scalar‘𝑤)) = (Base‘𝐹)) |
6 | asclfval.k | . . . . . 6 ⊢ 𝐾 = (Base‘𝐹) | |
7 | 5, 6 | eqtr4di 2793 | . . . . 5 ⊢ (𝑤 = 𝑊 → (Base‘(Scalar‘𝑤)) = 𝐾) |
8 | fveq2 6907 | . . . . . . 7 ⊢ (𝑤 = 𝑊 → ( ·𝑠 ‘𝑤) = ( ·𝑠 ‘𝑊)) | |
9 | asclfval.s | . . . . . . 7 ⊢ · = ( ·𝑠 ‘𝑊) | |
10 | 8, 9 | eqtr4di 2793 | . . . . . 6 ⊢ (𝑤 = 𝑊 → ( ·𝑠 ‘𝑤) = · ) |
11 | eqidd 2736 | . . . . . 6 ⊢ (𝑤 = 𝑊 → 𝑥 = 𝑥) | |
12 | fveq2 6907 | . . . . . . 7 ⊢ (𝑤 = 𝑊 → (1r‘𝑤) = (1r‘𝑊)) | |
13 | asclfval.o | . . . . . . 7 ⊢ 1 = (1r‘𝑊) | |
14 | 12, 13 | eqtr4di 2793 | . . . . . 6 ⊢ (𝑤 = 𝑊 → (1r‘𝑤) = 1 ) |
15 | 10, 11, 14 | oveq123d 7452 | . . . . 5 ⊢ (𝑤 = 𝑊 → (𝑥( ·𝑠 ‘𝑤)(1r‘𝑤)) = (𝑥 · 1 )) |
16 | 7, 15 | mpteq12dv 5239 | . . . 4 ⊢ (𝑤 = 𝑊 → (𝑥 ∈ (Base‘(Scalar‘𝑤)) ↦ (𝑥( ·𝑠 ‘𝑤)(1r‘𝑤))) = (𝑥 ∈ 𝐾 ↦ (𝑥 · 1 ))) |
17 | df-ascl 21893 | . . . 4 ⊢ algSc = (𝑤 ∈ V ↦ (𝑥 ∈ (Base‘(Scalar‘𝑤)) ↦ (𝑥( ·𝑠 ‘𝑤)(1r‘𝑤)))) | |
18 | 16, 17, 6 | mptfvmpt 7248 | . . 3 ⊢ (𝑊 ∈ V → (algSc‘𝑊) = (𝑥 ∈ 𝐾 ↦ (𝑥 · 1 ))) |
19 | fvprc 6899 | . . . . 5 ⊢ (¬ 𝑊 ∈ V → (algSc‘𝑊) = ∅) | |
20 | mpt0 6711 | . . . . 5 ⊢ (𝑥 ∈ ∅ ↦ (𝑥 · 1 )) = ∅ | |
21 | 19, 20 | eqtr4di 2793 | . . . 4 ⊢ (¬ 𝑊 ∈ V → (algSc‘𝑊) = (𝑥 ∈ ∅ ↦ (𝑥 · 1 ))) |
22 | fvprc 6899 | . . . . . . . . 9 ⊢ (¬ 𝑊 ∈ V → (Scalar‘𝑊) = ∅) | |
23 | 3, 22 | eqtrid 2787 | . . . . . . . 8 ⊢ (¬ 𝑊 ∈ V → 𝐹 = ∅) |
24 | 23 | fveq2d 6911 | . . . . . . 7 ⊢ (¬ 𝑊 ∈ V → (Base‘𝐹) = (Base‘∅)) |
25 | base0 17250 | . . . . . . 7 ⊢ ∅ = (Base‘∅) | |
26 | 24, 25 | eqtr4di 2793 | . . . . . 6 ⊢ (¬ 𝑊 ∈ V → (Base‘𝐹) = ∅) |
27 | 6, 26 | eqtrid 2787 | . . . . 5 ⊢ (¬ 𝑊 ∈ V → 𝐾 = ∅) |
28 | 27 | mpteq1d 5243 | . . . 4 ⊢ (¬ 𝑊 ∈ V → (𝑥 ∈ 𝐾 ↦ (𝑥 · 1 )) = (𝑥 ∈ ∅ ↦ (𝑥 · 1 ))) |
29 | 21, 28 | eqtr4d 2778 | . . 3 ⊢ (¬ 𝑊 ∈ V → (algSc‘𝑊) = (𝑥 ∈ 𝐾 ↦ (𝑥 · 1 ))) |
30 | 18, 29 | pm2.61i 182 | . 2 ⊢ (algSc‘𝑊) = (𝑥 ∈ 𝐾 ↦ (𝑥 · 1 )) |
31 | 1, 30 | eqtri 2763 | 1 ⊢ 𝐴 = (𝑥 ∈ 𝐾 ↦ (𝑥 · 1 )) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ∅c0 4339 ↦ cmpt 5231 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 Scalarcsca 17301 ·𝑠 cvsca 17302 1rcur 20199 algSccascl 21890 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-1cn 11211 ax-addcl 11213 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-nn 12265 df-slot 17216 df-ndx 17228 df-base 17246 df-ascl 21893 |
This theorem is referenced by: asclval 21918 asclfn 21919 asclf 21920 rnascl 21929 ressascl 21934 asclpropd 21935 rnasclg 42486 |
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